The former Celtic midfielder joins the Black Cats off the back of a four-year stay at Hibernian, with whom he won the Scottish Cup and the Scottish Championship title.
"Dylan has been really good for Hibernian and that's culminated in him earning international recognition," Sunderland boss Jack Ross told the club's official website.
"There were a lot of teams interested in him because he's at a good stage in his career and he's absolutely desperate to come down to England and show people how good a player he is.
"The fans will like him because he's got a lot of quality."
Sunderland have also completed the signing of left-back Reece James on a free transfer.